Integral Curves and Flows

Suppose is a smooth manifold. If is a smooth curve, then for each , the velocity vector is a vector in . In this section we describe a way to work backwards: given a tangent vector at each point, we seek a curve whose velocity at each point is equal to the given vector there.

ODEs on Manifolds

Integral Curve

An integral curve of a vector field is a smooth curve such that for all .

Theorem

Suppose is a smooth manifold. For any vector field and any , there exists a unique solution of the initial value problem for at . That is, there exists an integral curve of for some such that .

Proof In fact, this is a local problem at . We choose a chart at , say , then the initial value problem becomes a first order ODE in , which has a unique solution by the theorem.

e.g. Let and . Fix some initial point , choose as a chart, we solve the ODE to get for some constant . Now let , we obtain , so , which indicates that the maximal interval of the integral curve is .

Rescaling Lemma

Let be a smooth vector field on a smooth manifold , let be an interval, and let be an integral curve of . For any , the curve defined by is an integral curve of the vector field , where .

Translation Lemma

Let , , , and be as in the preceding lemma. For any , the curve defined by is also an integral curve of , where .

Another good property of solutions to initial value problems is that they depend smoothly on the initial point:

Vector Fields Generate Local Flows

For any vector field and any , there exists a neighbourhood of , such that if satisfies is an integral curve of at , then is a local diffeomorphism for all , and whenever .

Global Flows

An important special case is where the manifold is compact, or more generally where the support of the vector field is compact: In this case we have a globally defined flow.

Diffeomorphism Group of a Manifold

Let be a smooth manifold. Then, the diffeomorphism group of , denoted as , is the group of all diffeomorphisms with the operation of composition.

Global Flow

A global flow on is a smooth one parameter group of diffeomorphisms . This means that and for all , and is smooth on . In other words, it is a continuous left action of on .

Fundamental Theorem on Flows

Maximal Integral Curve & Maximal Flow

A maximal integral curve is one that cannot be extended to an integral curve on any larger open interval, and a maximal flow is a flow that admits no extension to a flow on a larger flow domain.

Fundamental Theorem on Flows

Let be a smooth vector field on a smooth manifold . There is a unique smooth maximal flow.

Complete Vector Field

Complete Vector Field

A vector field is complete if it generates a global flow on .

Compactly Supported Vector Fields Generate Global Flows

Suppose has compact support, that is, is a compact subset of . Then is complete.

Proposition

Every smooth vector field on a compact manifold without boundary (i.e. ) is complete. In general, if is a compact manifold with boundary, then every smooth vector field on is complete if and only if it is tangent to the boundary .

Commuting Vector Fields

Canonical Form for Commuting Vector Fields

Let be a smooth -manifold, and let be a linearly independent -tuple of smooth commuting vector fields on an open subset . For each , there exists a smooth coordinate chart centered at such that for .
If is an embedded codimension- submanifold and is a point of such that is complementary to the span of , then the coordinates can also be chosen such that is the slice defined by .

Lie Derivatives

Lie Derivative of Vector Fields

Suppose is a smooth manifold, and is a flow on . Let be a vector field defined by , then we define the Lie derivative of a vector field on as